Exercise 5, § 1

Show that if E is a Fréchet space and X is a locally compact space, then the space $\mathscr{K}(X; E)$ is bornological. From this, deduce that for every set $\mathscr{G}$ of bounded subsets of $\mathscr{K}(X; \mathbf{C})$ that is a covering of $\mathscr{K}(X; \mathbf{C})$ and is such that every set formed by the points of a convergent sequence in $\mathscr{K}(X; \mathbf{C})$ belongs to $\mathscr{G}$, the space $\mathscr{M}(X_\mathscr{G}; \mathbf{C})$ is complete.